We present recent work characterizing Ricci curvature and Ricci flow
on manifolds in terms of functional inequalities for heat semigroups.
The observation of Aaron Naber that bounded Ricci curvature on a
Riemannian manifold controls the analysis on path space, in a manner
analogous to how lower Ricci curvature bounds control the analysis
on the manifold, gave new impetus to the field. The functional inequalities allow in particular to characterize Einstein manifolds and Ricci solitons. The talk includes extensions of these methods to geometric flows on manifolds, as well as to the path space of Riemannian manifolds evolving under a geometric flow.